DOCSLIB.ORG

Melting of Two-Dimensional Tunable-Diameter Colloidal Crystals

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Melting of Two-Dimensional Tunable-Diameter Colloidal Crystals PHYSICAL REVIEW E 77, 041406 ͑2008͒ Melting of two-dimensional tunable-diameter colloidal crystals Y. Han,* N. Y. Ha,† A. M. Alsayed, and A. G. Yodh Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6396, USA ͑Received 12 February 2007; revised manuscript received 24 November 2007; published 18 April 2008͒ Melting of two-dimensional colloidal crystals is studied by video microscopy. The samples were composed of microgel spheres whose diameters could be temperature tuned, and whose pair potentials were measured to be short ranged and repulsive. We observed two-step melting from the crystal to a hexatic phase and from the hexatic to the liquid phase as a function of the temperature-tunable volume fraction. The translational and orientational susceptibilities enabled us to definitively determine the phase transition points, avoiding ambigu- ities inherent in other analyses and resolving a “dislocation precursor stage” in the solid phase that some of the traditional analyses may incorrectly associate with the hexatic phase. A prefreezing stage of the liquid with ordered patches was also found. DOI: 10.1103/PhysRevE.77.041406 PACS number͑s͒: 82.70.Ϫy, 64.60.Cn, 64.70.DϪ I. INTRODUCTION to follow the spatiotemporal evolution of the same sample area through the entire sequence of transitions. This feature ͑ ͒ Two-dimensional 2D melting is a classic problem in is attractive and was not realized in previous colloidal melt- ͓ ͔ condensed matter physics 1 , and, over the years, theories ing experiments which employed samples composed of attempting to understand 2D melting have emphasized topo- charged spheres in the wedge geometry with density gradi- ͓ ͔ ͓ ͔ logical defects 2–5 , geometrical defects 6 , and grain ents ͓11͔, nor in experiments that employed samples com- ͓ ͔ boundaries 7 . The most popular model for understanding posed of hard spheres in many different concentration- the transition is Kosterlitz-Thouless-Halperin-Nelson-Young dependent cells ͓12͔. The temperature-sensitive samples ͑ ͒ ͓ ͔ KTHNY theory 2–5 , which predicts two-step melting, employed herein start in the equilibrium crystal phase and from the crystal to a hexatic phase and then from the hexatic reequilibrate rapidly after each tiny temperature jump. There- to a liquid phase. The intermediate hexatic phase has short- fore, they are unlikely to be trapped in metastable glassy range translational and quasi-long-range orientational order, states during melting. Beautiful recent experiments using and the two transitions are beautifully characterized by the magnetic spheres with tunable dipole-dipole interactions creation, binding, and unbinding of topological defects, i.e., ͓13–15͔ share some of these advantages but also differ from dislocations and disclinations, respectively. Experimenters the present experiments in a complementary way as a result have sought out these features across a wide range of mate- of their long-range dipolar interactions. ͓ ͔ rials, including monolayers of molecules and electrons 1 , We measured a variety of sample properties during melt- ͓ ͔ ͓ ͔ liquid crystals 8 , superconductors 9 , diblock copolymers ing, including radial distribution functions, structure factors, ͓ ͔ ͓ ͔ 10 , and colloidal suspensions 11–16 . Some experiments topological defect densities, dynamic Lindemann parameters and simulations have demonstrated substantial agreement ͓13͔, translational and orientational order parameters, and or- with KTHNY theory, but others exhibit deviations and am- der parameter correlation functions in space and time. In this ͓ ͔ biguities possibly due to finite-size effects 1 , the interaction process we discovered that the order parameter susceptibility, ͓ ͔ ͓ ͔ range and form 16 , and out-of-plane fluctuations 17,18 .In i.e., the order parameter fluctuations, proved superior for our view, the collection of evidence clearly points to the finding phase transition points compared to other analyses validity of the KTHNY scenario in 2D systems with long- which typically suffer finite-size and/or finite-time ambigu- ͓ ͔ range interaction potentials 13–15,19 , but the evidence is ities. The susceptibility method has been applied in simula- less convincing in systems with short-range interactions tions ͓17,21͔, but to our knowledge has not been applied in ͓ ͔ 20–24 . Consequently it remains desirable to explore the imaging experiments. Using this method, we clearly resolved phenomena in other model systems, especially those with the intermediate hexatic phase, and we identified a “disloca- short-ranged interactions. tion precursor stage” in the crystal phase that traditional In this paper we examine 2D melting in a colloidal sys- analyses sometimes incorrectly assign to the hexatic phase. tem. The pair potential between particles in this colloidal In addition, the functional form of the susceptibility near the suspension is short ranged and repulsive, and the sphere di- phase transition points, e.g., divergences or discontinuities, ameter is thermally sensitive, so that temperature tuning can can be used to determine the order of the phase transition. be used to vary the sample volume fraction and drive the melting transition. Temperature-sensitive particles enable us II. EXPERIMENT A. Sample preparation and characterization *Present address: Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong. The samples consisted of a monolayer of N-isopropyl †Present address: Department of Physics, Ajou University, Suwon acrylamide ͑NIPA͒ spheres confined between two glass cov- 443-749, Korea. erslips. We synthesized NIPA microgel spheres by free- 1539-3755/2008/77͑4͒/041406͑7͒ 041406-1 ©2008 The American Physical Society HAN et al. PHYSICAL REVIEW E 77, 041406 ͑2008͒ radical polymerization ͓25͔ and suspended them in buffer 4 (A) ρ=0.909 (26.9°C) (B) solution ͑pH=4.0, 20 mM acetic acid͒. The NIPA polymer ° becomes less hydrophilic at high temperature ͓26͔, and there- 3 30 C ° T] 24 C fore the sphere diameter decreases with increasing tempera- B ture as water moves out of the microgel particle. Dynamic [k ) 2 r light scattering measurements found the NIPA sphere hydro- ( dynamic diameter to vary linearly from 950 nm at 20 °C to u 740 nm at 30 °C and showed that sphere polydispersity was 1 ↓ less than 3%. Such small polydispersity should not affect the ͓ ͔ nature of the melting process 27 . The cleaned glass sur- 0 ↑ faces were coated with a layer of 100-nm-diameter NIPA 0.5 1r[µm] 1.5 spheres to prevent particle sticking. A simple geometrical ρ=0.890 (27.5°C) (C) ρ=0.858 (27.9°C) (D) calculation showed that 100 nm close packed spheres on the surface give rise to ϳ3 nm surface roughness for the 800- nm-diameter spheres. This surface roughness is negligible compared to sphere polydispersity and wall separation fluc- tuations. In addition, our observations of the large sphere motions at lower concentrations did not find evidence of preferential spatial locations, i.e., of significant surface po- tentials. The dense monolayer of 800 nm spheres formed crystal domains within the sample cell of typical size ϳ͑40 ␮m͒2, ͑ ͒ ͑ ͒ ͑ ͒ corresponding to ϳ3000 particles. Measurements were car- FIG. 1. a Pair potential u r of NIPA spheres at 24 squares ͑ ͒ ried out on a ϳ͑20 ␮m͒2 central area away from the grain and 30 °C circles . Arrows indicate corresponding hydrodynamic ͑ ͒ ͑ ͒ boundaries. In practice we found that grain boundaries af- radius measured by dynamic light scattering. b – d : Typical 10 s fected only a few neighboring lines of particles, and that particle trajectories in the crystal, hexatic, and liquid phases, respectively. melting started from both inside crystal domains and at grain boundaries. This behavior differs qualitatively from grain- boundary melting in 3D ͓25͔ and edge melting in 2D ͓28͔, temperature using the method described in Ref. ͓31͔. From wherein melting starts from grain boundaries or edges and g͑r͒, we applied liquid structure theory to extract ͓32͔ the then propagates into the crystal. Our observations suggest pair potentials u͑r͒ shown in Fig. 1. Note that the potentials that interfacial energies for liquid nucleation from anywhere are short ranged and repulsive. The effective particle diam- ϳ within the crystal are similar to those near grain boundaries. eter at 1kBT is 10% smaller than the hydrodynamic diam- It appears that melting starts nearly simultaneously through- eter measured by dynamic light scattering. Herein we use the out the crystal. hydrodynamic diameter ␴ for defining the areal density ␳ 2 The sample was heated very slowly in 0.1 °C steps for a =n␲␴ /4, where n is the areal number density. few minutes, and the measurements were taken after the tem- perature stabilized. We did not observe convection, even in B. Order parameter correlations in space and time samples with smaller particles at much lower volume frac- Figures 1͑b͒–1͑d͒ show typical particle trajectories in the tions. The sample cell was approximately 1 ␮m thick, much three phases. In KTHNY theory, the traditional way to dis- smaller than the 1-mm-thick glass slide on which it was tinguish phases derives from the shapes of the order param- mounted. The temperature difference across the slide was eter correlation functions. For our analysis we first labeled ͑ ␺ ␺ ͒ ␺ about 5 °C, so the gradient across the sample was very small each particle by xj ,yj ,t, 6j , Tj . Here t is time, 6j ␪ ͑Ͻ ͒ ͚͑nn 6i jk͒/N 0.01 °C . Furthermore, the Reynolds number of water at = k=1e is the orientational order parameter, and ␮ ␺ iG·rj this 1 m scale is small, making convection unlikely. Be- Tj=e is the translational order parameter for particle j at ͑ ͒ ␪ fore starting the measurements we cycled the samples once position rj = xj ,yj . jk is the angle of the bond between or twice above the melting point in order to relieve any pos- particle j and its neighbor k.
Load more

Recommended publications
  • Bond-Orientational Order and Frank's Constant in Two-Dimensional
    IOP Journal of Physics: Condensed Matter Journal of Physics: Condensed Matter J. Phys.: Condens. Matter J. Phys.: Condens. Matter 30 (2018) 104003 (6pp) https://doi.org/10.1088/1361-648X/aaab31 30 Bond-orientational order and Frank’s 2018 constant in two-dimensional colloidal © 2018 IOP Publishing Ltd hard spheres JCOMEL Alice L Thorneywork1,2, Joshua L Abbott1, Dirk G A L Aarts1, Peter Keim3 and Roel P A Dullens1 104003 1 Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford, OX1 3QZ, United Kingdom A L Thorneywork et al 2 Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom 3 Department of Physics, University of Konstanz, Konstanz 78464, Germany Bond-orientational order and Frank s constant in two-dimensional colloidal hard spheres ’ E-mail: [email protected] Printed in the UK Received 26 October 2017, revised 12 January 2018 Accepted for publication 29 January 2018 Published 15 February 2018 CM Abstract Recently, the full phase behaviour of 2D colloidal hard spheres was experimentally 10.1088/1361-648X/aaab31 established, and found to involve a first order liquid to hexatic transition and a continuous hexatic to crystal transition (Thorneywork et al 2017 Phys. Rev. Lett. 118 158001). Here, we expand upon this work by considering the behaviour of the bond-orientational correlation Paper time and Frank’s constant in the region of these phase transitions. We also consider the excess entropy, as calculated from the radial distribution functions, for a wide range of area fractions covering the liquid, hexatic and crystal phases.
    [Show full text]
  • Kosterlitz–Thouless Physics: a Review of Key Issues Accepted for Publication 7 October 2015 Published 28 January 2016
    Reports on Progress in Physics REVIEW Related content - Berezinskii–Kosterlitz–Thouless transition Kosterlitz–Thouless physics: a review of key and two-dimensional melting V N Ryzhov, E E Tareyeva, Yu D Fomin et issues al. - Depinning and nonequilibrium dynamic phases of particle assemblies driven over To cite this article: J Michael Kosterlitz 2016 Rep. Prog. Phys. 79 026001 random and ordered substrates: a review C Reichhardt and C J Olson Reichhardt - Quantum phase transitions Matthias Vojta View the article online for updates and enhancements. Recent citations - Isolating long-wavelength fluctuation from structural relaxation in two-dimensional glass: cage-relative displacement Hayato Shiba et al - Disappearance of the Hexatic Phase in a Binary Mixture of Hard Disks John Russo and Nigel B. Wilding - Berezinskii—Kosterlitz—Thouless transition and two-dimensional melting Valentin N. Ryzhov et al This content was downloaded from IP address 128.227.24.141 on 25/02/2018 at 11:47 IOP Reports on Progress in Physics Reports on Progress in Physics Rep. Prog. Phys. Rep. Prog. Phys. 79 (2016) 026001 (59pp) doi:10.1088/0034-4885/79/2/026001 79 Review 2016 Kosterlitz–Thouless physics: a review of key © 2016 IOP Publishing Ltd issues RPPHAG J Michael Kosterlitz 026001 Department of Physics, Brown University, Providence, RI 02912, USA J M Kosterlitz E-mail: [email protected] Received 1 June 2015, revised 5 October 2015 Kosterlitz–Thouless physics: a review of key issues Accepted for publication 7 October 2015 Published 28 January 2016 Printed in the UK Abstract This article reviews, from a very personal point of view, the origins and the early work on ROP transitions driven by topological defects such as vortices in the two dimensional planar rotor model and in 4Helium films and dislocations and disclinations in 2D crystals.
    [Show full text]
  • Frank's Constant in the Hexatic Phase
    PHYSICAL REVIEW E 75, 031402 ͑2007͒ Frank’s constant in the hexatic phase P. Keim,1 G. Maret,2 and H. H. von Grünberg1 1Universität Graz, 8010 Graz, Austria 2Universität Konstanz, 78457 Konstanz, Germany ͑Received 1 September 2006; revised manuscript received 25 January 2007; published 22 March 2007͒ Using videomicroscopy data of a two-dimensional colloidal system the bond-order correlation function G6 is ␰ calculated and used to determine both the orientational correlation length 6 in the liquid phase and the modulus of orientational stiffness, Frank’s constant FA, in the hexatic phase. The latter is an anisotropic fluid phase between the crystalline and the isotropic liquid phase. FA is found to be finite within the hexatic phase, takes the value 72/␲ at the hexatic↔isotropic liquid phase transition, and diverges at the hexatic↔crystal transition as predicted by the Kosterlitz-Thouless-Halperin-Nelson-Young theory. This is a quantitative test of the mechanism of breaking the orientational symmetry by disclination unbinding. DOI: 10.1103/PhysRevE.75.031402 PACS number͑s͒: 82.70.Dd, 64.70.Dv, 68.35.Rh I. INTRODUCTION type. Defect dissociation is completed at a temperature where the thermally averaged pair distance The theory of melting in two dimensions ͑2D͒, developed ␤ ͐ 2 2 − Hd by Kosterlitz, Thouless, Halperin, Nelson, and Young 2 d rr e 2−cd 2 ͗r ͘ = ␤ = r ͑2͒ ͑KTHNY͒ has been a matter of debate over decades. In a 2D ͐ 2 − Hd c d re 4−cd crystal, the density-density correlation function is an algebra- → ically decaying function and is not given by a set of ␦ peaks diverges which is obviously the case if cd 4.
    [Show full text]
  • Berezinskii-Kosterlitz-Thouless Transition
    Melting scenarios and unusual crystal structures in two-dimensional soft core systems Valentin N. Ryzhov, Yury D. Fomin, Elena E. Tareyeva, Elena N. Tsiok Institute for High Pressure Physics, Russian Academy of Sciences, 108840 Troitsk, Moscow, Russia Motivation and Outline • Dependence of structural properties and melting scenarios on the shape of the core softened and bound potentials in 2D • Influence of random pinning on the melting scenario of core-softened and bound potential systems • Theoretical background: Berezinskii-Kosterlitz-Thouless transition • Core softened and bound potentials • Theoretical background: Melting scenarios in 2D • Phase diagrams of core-softened and bound potential systems and influence of random pinning on the melting of these systems Computer simulations and experimental study of two-dimensional (2D) systems For simple potentials (hard disks, soft disks, Lennard-Jones, etc.) ground state - closed packed triangle crystal structure: Computer simulations and experimental study of water in slit pores Monolayer ice: simulation results for the TIP5P model of water in a quasi-two-dimensional hydrophobic slit nanopore (Ronen Zangi* and Alan E. Mark, Phys. Rev. Lett. 91, 025502 (2003); P. Kumar, S. V. Buldyrev, F. W. Starr, N. Giovambattista, and H. Eugene Stanley, Phys. Rev. E 72, 051503 (2005)) . First-order transition into square phase. Computer simulations and experimental study of water in slit pores The phase diagram of two layers of (TIP4P and mW) water confined between parallel non hydrogen bonding walls (Jessica C. Johnston, Noah Kastelowitz, and Valeria Molinero, THE JOURNAL OF CHEMICAL PHYSICS 133, 154516 (2010)). Computer simulations and experimental study of water in slit pores The nanoconfined between two graphene sheets water at room temperature forms ‘square ice’- a phase having symmetry qualitatively different from the conventional tetrahedral geometry of hydrogen bonding between water molecules.
    [Show full text]
  • Existence of a Hexatic Phase in Porous Media
    VOLUME 89, NUMBER 7 PHYSICAL REVIEW LETTERS 12AUGUST 2002 Existence of a Hexatic Phase in Porous Media Ravi Radhakrishnan,1 Keith E. Gubbins,2 and Malgorzata Sliwinska-Bartkowiak3 1Massachusetts Institute of Technology, 77 Massachusetts Avenue, 66-021, Cambridge, Massachusetts 02139 2North Carolina State University, 113 Riddick Labs, Raleigh, North Carolina 27695 3Institute of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland (Received 7 August 2000; revised manuscript received 17 April 2002; published 25 July 2002) Molecular simulations for simple fluids in narrow slit-shaped carbon pores exhibit crystal-hexatic and hexatic-liquid transitions that are consistent with Kosterlitz-Thouless-Halperin-Nelson-Young theory. The temperature range over which the hexatic phase is stable is dramatically widened under confinement. Remarkably, the transitions, which are continuous for a single adsorbed layer, become weakly first order when the pore can accommodate two molecular layers. Nonlinear dielectric effect measurements for CCl4 and aniline in activated carbon fibers (pore width 1.4 nm) show divergence at these transitions, confirming the hexatic phase. DOI: 10.1103/PhysRevLett.89.076101 PACS numbers: 68.35.Rh Two-dimensional systems have a special significance for ment of thermal equilibrium because of large particle sizes phase transitions in which continuous symmetry is broken compared to molecular dimensions. (such as freezing transitions). The Mermin-Wagner theo- Activated carbon fibers (ACF) possess microcrystallites rem states that true long-range order cannot exist in such made up of graphene sheets that tend to align in similar systems [1]. Nelson and Halperin [2] proposed the directions, with slit-shaped voids between the microcrys- KTHNY (Kosterlitz-Thouless-Halperin-Nelson-Young) tals.
    [Show full text]
  • Experimental Test of Two-Dimensional Melting Through Disclination Unbinding
    PHYSICAL REVIEW E, VOLUME 64, 051404 Experimental test of two-dimensional melting through disclination unbinding R. A. Quinn* and J. Goree† Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242 ͑Received 8 November 2000; revised manuscript received 30 May 2001; published 29 October 2001͒ A two-dimensional ͑2D͒ melting transition has been studied in a nonequilibrium experimental model system. The system used was a complex or dusty plasma consisting of microspheres suspended in a glow-discharge plasma, where we have mapped the topological defects during the transition. The role of the defects in the melting transition is evaluated and the arrangement of the defects in the lattice is quantified in a new way. It is found that defect density increases dramatically during the melting; at all stages the defects tend to be clustered together rather than widely dispersed; the clustering tends to take the form of chain or string-like structures. We compare these results for the defect structure with the assumptions of the popular 2D melting theory of Halperin and Nelson, rather than the predictions, as is more common. DOI: 10.1103/PhysRevE.64.051404 PACS number͑s͒: 82.70.Dd, 64.60.Ϫi, 52.27.Lw, 68.35.Rh I. INTRODUCTION rections, and the lattice microstructure is easily detected by Experimental model systems have played an important direct imaging. However, complex plasmas differ from col- role in understanding the nature of two-dimensional ͑2D͒ loidal suspensions in their volume fraction and equilibration melting for more than a decade. Much of the current interest time, which are both smaller by a factor as large as 105.
    [Show full text]
  • Early Work on Defect Driven Phase Transitions
    May 13, 2013 11:19 8572: 40 Years of BKT Theory chap1 Chapter 1 Early Work on Defect Driven Phase Transitions J. Michael Kosterlitz Department of Physics, Brown University, Providence, RI 02912, USA David J. Thouless Department of Physics, Box 351560, University of Washington, Seattle, WA 98195, USA This article summarizes the early history of the theory of phase transitions driven by topological defects, such as vortices in superfluid helium films or dislocations and disclinations in two-dimensional solids. We start with a review of our two earliest papers, pointing out their errors and omis- sions as well as their insights. We then describe the work, partly done by Kosterlitz but mostly done by other people, which corrected these over- sights, and applied these ideas to experimental systems, and to numerical and experimental simulations. by WSPC on 11/13/13. For personal use only. 1.1. Introduction The idea that certain phase transitions might be driven by the occurrence of an equilibrium concentration of defects such as dislocations in solids or 40 Years of Berezinskii–Kosterlitz–Thouless Theory Downloaded from www.worldscientific.com quantized vortices in superfluids is almost as old as the recognition that such defects have an important role in the mechanical or electrical prop- erties of such materials. Indeed, in Onsager’s incredibly brief initial note on the existence of quantized vortices in superfluid helium, the possibility of a vortex-driven transition from the superfluid phase to the normal fluid phase is mentioned,1 and this possibility was later presented by Feynman2 in a much more accessible form.
    [Show full text]
  • Phases of Active Matter in Two Dimensions
    Phases of active matter in two dimensions Leticia F. Cugliandolo Sorbonne Universit´e, Laboratoire de Physique Th´eoriqueet Hautes Energies, CNRS UMR 7589 Tour 13, 5`eme´etage,4 Place Jussieu, 75252 Paris Cedex 05, France Giuseppe Gonnella Universit`adegli Studi di Bari and INFN Sezione di Bari, Dipartimento di Fisica, via Amendola 173, I-70126 Bari, Italia arXiv:1810.11833v1 [cond-mat.stat-mech] 28 Oct 2018 iv 1 v Preface These notes focus on the description of the phases of matter in two dimensions. Firstly, we present a brief discussion of the phase diagrams of bidimensional interacting pas- sive systems, and their numerical and experimental measurements. The presentation will be short and schematic. We will complement these notes with a rather complete bibliography that should guide the students in their study of the development of this very rich subject over the last century. Secondly, we summarise very recent results on the phase diagrams of active Brownian disks and active dumbbell systems in two dimensions. The idea is to identify all the phases and to relate, when this is possible, the ones found in the passive limit with the ones observed at large values of the ac- tivity, at high and low densities, and for both types of constituents. Proposals for the mechanisms leading to these phases will be discussed. The physics of bidimensional active systems open many questions, some of which will be listed by the end of the Chapter. Acknowledgements We want to warmly thank our collaborators on studies of active matter; in alphabeti- cal order Mathias Casiulis, Olivier Dauchot, Pasquale Digregorio, Gianluca Laghezza, Antonio Lamura, Demian Levis, Davide Loi, Davide Marenduzzo, Alessandro Mossa, Stefano Mossa, Ignacio Pagonabarraga, Isabella Petrelli, Antonio Suma, and Marco Tarzia, from whom we learnt most of what we present in these notes.
    [Show full text]
  • Arxiv:2007.00297V1 [Cond-Mat.Soft] 1 Jul 2020 Disclination Unbinding for Hexatic- fluid Melting
    Two step melting of the Weeks- Chandler- Anderson system in two dimensions Shubhendu Shekhar Khali,1, ∗ Dipanjan Chakraborty,1, y and Debasish Chaudhuri2, 3, z 1Department of Physical Science, Indian Institute Of Science Education and Research Mohali, Punjab, India 140306 2Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India 3Homi Bhaba National Institute, Anushaktigar, Mumbai 400094, India (Dated: July 2, 2020) We present a detailed numerical simulation study of a two dimensional system of particles in- teracting via the Weeks- Chandler- Anderson potential, the repulsive part of the Lennard- Jones potential. With reduction of density, the system shows a two- step melting: a continuous melting from solid to hexatic phase, followed by a a first order melting of hexatic to liquid. The solid- hexatic melting is consistent with the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) scenario and shows dislocation unbinding. The first order melting of hexatic to fluid phase, on the other hand, is dominated by formation of string of defects at the hexatic- fluid interfaces. I. INTRODUCTION a continuous melting transition, and a first order tran- sition [17]. This simulation did not allow the formation The high density phase of mono-dispersed particles in of defects but counted all the Monte-Carlo moves that two dimensions (2d) can not support long ranged posi- potentially formed them, to calculate the defect- core- tional order due to thermal fluctuations [1{3]. However, energy, and the unrenormalized Young's modulus. These the pair correlation function in it does not decay expo- two quantities along with the KTHNY recursion relation nentially, unlike in the fluid phase.
    [Show full text]
  • Two-Stage Melting Induced by Dislocations and Grain Boundaries in Monolayers of Hard Spheres
    Two-stage melting induced by dislocations and grain boundaries in monolayers of hard spheres Weikai Qi, Anjan P. Gantapara, and Marjolein Dijkstra∗ Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands (Dated: January 28, 2014) Melting in two-dimensional systems has remained controversial as theory, simulations, and experi- ments show contrasting results. One issue that obscures this discussion is whether or not theoretical predictions on strictly 2D systems describe those of quasi-2D experimental systems, where out- of-plane fluctuations may alter the melting mechanism. Using event-driven Molecular Dynamics simulations, we find that the peculiar two-stage melting scenario of a continuous solid-hexatic and a first-order hexatic-liquid transition as observed for a truly 2D system of hard disks [Bernard and Krauth, Phys. Rev. Lett. 107, 155704 (2011)] persists for a quasi-2D system of hard spheres with out-of-plane particle motions as high as half the particle diameter. By calculating the renormalized Young's modulus, we show that the solid-hexatic transition is of the Kosterlitz-Thouless type, and occurs via dissociation of bound dislocation pairs. In addition, we find a first-order hexatic-liquid transition that seems to be driven by a spontaneous proliferation of grain boundaries. PACS numbers: 82.70.Dd,64.70.D-,61.20.Ja I. INTRODUCTION first-order melting transition [10{22]. These results seem to suggest that 2D melting is not universal, but depends Melting in two-dimensional (2D) systems has been de- on specific properties of the system, e.g., interparticle po- bated heavily since Landau, Peierls, and Mermin showed tential, out-of-plane fluctuations, finite-size effects, etc.
    [Show full text]
  • Arxiv:1804.05582V5 [Cond-Mat.Soft] 28 May 2019
    Melting and re-entrant melting of polydisperse hard disks Pablo Sampedro Ruiz,1 Qun-li Lei,1 and Ran Ni1, ∗ 1School of Chemical and Biomedical Engineering, Nanyang Technological University, 62 Nanyang Drive, 637459, Singapore Abstract Because of long-wavelength fluctuations, the nature of solids and phase transitions in 2D are different from those in 3D systems, and have been heavily debated in past decades, in which the focus was on the existence of hexatic phase. Here, by using large scale computer simulations, we investigate the melting transition in 2D systems of polydisperse hard disks. We find that, with increasing the particle size polydispersity, the melting transition can be qualitatively changed from the recently proposed two-stage process to the Kosterlitz-Thouless-Halperin-Nelson-Young scenario with significantly enlarged stability range for hexatic phase. Moreover, re-entrant melting transi- tions are found in high density systems of polydisperse hard disks, which were proven impossible in 3D polydisperse hard-sphere systems. These suggest a new fundamental difference between phase transitions in polydisperse systems in 2D and 3D. arXiv:1804.05582v5 [cond-mat.soft] 28 May 2019 ∗ [email protected] 1 I. INTRODUCTION Melting of two-dimensional solids has been heavily discussed since it was proven that long- wavelength thermal fluctuations prevent the long-range positional order in 2D systems [1{3]. A significant attempt to reach the general description of 2D melting was the Kosterlitz- Thouless-Halperin-Nelson-Young (KTHNY) theory, that predicts a new phase of matter, i.e., the hexatic phase, which has quasi-long-range orientational order and short-range positional order.
    [Show full text]
  • Melting of Crystals in Two Dimensions Urs Gasser,*[A] Christoph Eisenmann,[B] Georg Maret,[B] and Peter Keim[B]
    DOI: 10.1002/cphc.200900755 Melting of Crystals in Two Dimensions Urs Gasser,*[a] Christoph Eisenmann,[b] Georg Maret,[b] and Peter Keim[b] While the melting of crystals is in general not understood in conclusive evidence for the validity of this theory on a micro- detail on a microscopic scale, there is a microscopic theory for scopic level. Furthermore, we show how the mechanism of a class of two-dimensional crystals, which is based on the for- melting depends on the particle interaction and that a strong mation and unbinding of topological defects. Herein, we anisotropy of the interaction leads to a changed melting sce- review experimental work on a colloidal two-dimensional nario. model system with tunable interactions that has given the first 1. Introduction Like other phase transitions, the melting of crystals is a central 3D. This was first shown by Peierls for the magnetic XY- topic in condensed-matter physics and is of interest for the un- system.[7] In a ferromagnetic material, the tilting of a spin by a derstanding of all crystalline materials. Freezing and melting small angle df with respect to its neighbors takes an energy have been studied for centuries, which reflects their broad (E / d2). The excitation of a spin-wave with wavelength L relevance for many fields. Nevertheless, a detailed microscopic then requires an energy / Ldd2 / Ldð2p=LÞ2 , where d2{1,2,3} understanding of the melting transition is lacking for most ma- is the dimensionality. This energy diverges with L for d =3, terials, although a microscopic theory for the melting of crys- while for d =2 it is independent of L and for d= 1 it decreases tals in two dimensions (2D) has been developed in the 1970s / LÀ1.
    [Show full text]